Let $K$ be a $p$-adic local field, and $L$ the Lubin-Tate extension obtained from $K$ by attaching roots of some Lubin-Tate formal $\mathcal{O}_{K}$-module with $Gal(L/K) \simeq \mathcal{O}_{K}^{\times}$. Let $L'$ be the $p$-adic completion of the latter.
I think it follows from Tate's work in his paper on $p$-divisible groups that
$H_{c}^{i}(Gal(L, K), L') \simeq K$
for $i = 0, 1$ and that the other cohomology groups vanish. Here, this is continuous cohomology where the Galois group has its profinite topology and $L'$ has the $p$-adic topology.
What about the groups $H^{i}_{c}(Gal(L/K), \mathcal{O}_{L'})$? Are these known integrally, perhaps for some specific choices of $K$? For example, what if $K = \mathbb{Q}_{p}$ and $L$ is the cyclotomic extension attaching all $p$-th roots of unity?
